Stephens' Constant

Let  and  be nonzero integers such that  (except when ). Also let  be the set of primes  for which  for some nonnegative integer . Then assuming the generalized Riemann hypothesis, Stephens (1976) showed that the density of  relative to the primes is a rational multiple of

(OEIS A065478), where  is the th prime (Finch 2003).

REFERENCES:

Finch, S. R. "Artin's Constant." §2.4 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 104-110, 2003.

Moree, P. "Approximation of Singular Series and Automata." Submitted to Manuscripta Math. 101, 385-399, 2000.

Moree, P. and Stevenhagen, P. "A Two-Variable Artin Conjecture." J. Number Th. 85, 291-304, 2000.

Niklasch, G. "Some Number-Theoretical Constants." http://www.gn-50uma.de/alula/essays/Moree/Moree.en.shtml.

Sloane, N. J. A. Sequence A065478 in "The On-Line Encyclopedia of Integer Sequences."

Stephens, P. J. "Prime Divisor of Second-Order Linear Recurrences, I." J. Number Th. 8, 313-332, 1976.